The Markov approximation of the sequences of N-valued random variables and a class of small deviation theorems

摘要

Let {X-n, n greater than or equal to 0) be a sequence of random variables on the probability space (Omega, F, P) taking values in the alphabet S = {1, 2,.... N), and Q be another probability measure on F, under which {X-n, n greater than or equal to 0) is a Markov chain. Let h(P Q) be the sample divergence rate of P with respect to Q related to {X-n}. In this paper the Markov approximation of {X-n, n greater than or equal to 0} under P is discussed by using the notion of h(P Q), and a class of small deviation theorems for the averages of the bivariate functions of {X-n, n greater than or equal to 0} are obtained. In the proof an analytic technique in the study of a.c. convergence together with the martingale convergence theorem is applied. (C) 2000 Elsevier Science B.V. All rights reserved. MSC: primary 94A17; secondary 60F15. [References: 17]